常微分方程基礎(chǔ)理論（影印版）

前言

為了更好地借鑒國外數(shù)學(xué)教育與研究的成功經(jīng)驗，促進我國數(shù)學(xué)教育與研究事業(yè)的發(fā)展，提高高等學(xué)校數(shù)學(xué)教育教學(xué)質(zhì)量，本著“為我國熱愛數(shù)學(xué)的青年創(chuàng)造一個較好的學(xué)習(xí)數(shù)學(xué)的環(huán)境”這一宗旨，天元基金贊助出版“天元基金影印數(shù)學(xué)叢書”。該叢書主要包含國外反映近代數(shù)學(xué)發(fā)展的純數(shù)學(xué)與應(yīng)用數(shù)學(xué)方面的優(yōu)秀書籍，天元基金邀請國內(nèi)各個方向的知名數(shù)學(xué)家參與選題的工作，經(jīng)專家遴選、推薦，由高等教育出版社影印出版。為了提高我國數(shù)學(xué)研究生教學(xué)的水平，暫把選書的目標(biāo)確定在研究生教材上。當(dāng)然，有的書也可作為高年級本科生教材或參考書，有的書則介于研究生教材與專著之間。歡迎各方專家、讀者對本叢書的選題、印刷、銷售等工作提出批評和建議。

內(nèi)容概要

　　本書為開展常微分方程研究工作的讀者提供必要的準(zhǔn)備知識，可作為本科高年級和研究生常微分方程課程教材。　　本書內(nèi)容分為四部分：第一部分(第一、二、三章)的內(nèi)容包括解的存在性、唯一性、對數(shù)據(jù)的光滑依賴性，以及解的非唯一性；第二部分(第四、六、七章)討論線性常微分方程，書中用矩陣的S-N分解代替Jordan分解，前者的計算較后者更容易；第三部分(第八、九、十章)討論非線性常微分方程的穩(wěn)定性、漸近穩(wěn)定性等幾何理論；第四部分(第五、十一，十二、十三章)討論常微分方程的冪級數(shù)解，包括線性常微分方程的奇點分類及非線性常微分方程當(dāng)參數(shù)或自變量趨向某奇點時的漸近解等。

書籍目錄

PrefaceChapter Ⅰ.Fundamental Theorems of Ordinary Differential Equations?、?1.Existence and uniqueness with the Lipschitz condition?、?2.Existence without the Lipschitz condition?、?3.Some global properties of solutions　Ⅰ-4.Analytic differential equations　Exercises ⅠChapterⅡ.Dependence on Data?、?1.Continuity with respect to initial data and parameters　Ⅱ-2.Differentiability　Exercises ⅡChapter Ⅲ.Nonuniqueness?、?1.Examples?、?2.The Kneser theorem?、?3.Solution curves on the boundary of R(A)?、?4.Maximal and minimal solutions?、?5.A comparison theorem　Ⅲ-6.Sufficient conditions for uniqueness　Exercises ⅢChapter Ⅳ.General Theory of Linear Systems?、?1.Some basic results concerning matrices?、?2.Homogeneous systems of linear differential equations　Ⅳ-3.Homogeneous systems with constant coefficients?、?4.Systems with periodic coefficients　Ⅳ-5.Linear Hamiltonian systems with periodic coefficients?、?6.Nonhomogeneous equations?、?7.Higher-order scalar equations　Exercises ⅣChapter Ⅴ.Singularities of the First Kind?、?1.Formal solutions of an algebraic differential equation　Ⅴ-2.Convergence of formal solutions of a system of the first kind?、?3.The S-N decomposition of a matrix of infinite order　Ⅴ-4.The S-N decomposition of a differential operator?、?5.A normal form of a differential operator　Ⅴ-6.Calculation of the normal form of a differential operator?、?7.Classification of singularities of homogeneous linear systems　Exercises ⅤChapter Ⅵ.Boundary-Value Problems of Linear Differential Equations of the Second-Order?、?1.Zeros of solutions?、?2.Sturm-Liouville problems?、?3.Eigenvalue problems?、?4.Eigenfunction expansions?、?5.Jost solutions?、?6.Scattering data?、?7.Refiectionless potentials?、?8.Construction of a potential for given data?、?9.Differential equations satisfied by reflectionless potentials　Ⅵ-10.Periodic potentials　Exercises ⅥChapter Ⅶ.Asymptotic Behavior of Solutions of Linear Systems?、?1.Liapounoff's type numbers　Ⅶ-2.Liapounoff's type numbers of a homogeneous linear system?、?3.Calculation of Liapounoff's type numbers of solutions?、?4.A diagonalization theorem?、?5.Systems with asymptotically constant coefficients?、?6.An application of the Floquet theorem　Exercises ⅦChapter Ⅷ.Stability　Ⅷ-1.Basic definitions?、?2.A sufficient condition for asymptotic stability　Ⅷ-3.Stable manifolds?、?4.Analytic structure of stable manifolds?、?5.Two-dimensional linear systems with constant coefficients?、?6.Analytic systems in R2　Ⅷ-7.Perturbations of an improper node and a saddle point?、?8.Perturbations of a proper node　Ⅷ-9.Perturbation of a spiral point?、?10.Perturbation of a center　Exercises ⅧChapter Ⅸ.Autonomous Systems?、?1.Limit-invariant sets?、?2.Liapounoff's direct method　Ⅸ-3.Orbital stability?、?4.The Poincare-Bendixson theorem　Ⅸ-5.Indices of Jordan curves　Exercises ⅨChapter Ⅹ.The Second-Order Differential Equation (d2x)/(dt2)+h(x)*(dx)/(dt)+g(x)=0?、?1.Two-point boundary-value problems?、?2.Applications of the Liapounoff functions?、?3.Existence and uniqueness of periodic orbits?、?4.Multipliers of the periodic orbit of the van der Pol equation　Ⅹ-5.The van der Pol equation for a small ε > 0?、?6.The van der Pol equation for a large parameter?、?7.A theorem due to M.Nagumo　Ⅹ-8.A singular perturbation problem　Exercises ⅩChapter Ⅺ.Asymptotic Expansions?、?1.Asymptotic expansions in the sense of Poincare?、?2.Gevrey asymptotics　Ⅺ-3.Flat functions in the Gevrey asymptotics?、?4.Basic properties of Gevrey asymptotic expansions　Ⅺ-5.Proof of Lemma Ⅺ-2-6　Exercises ⅪChapter Ⅻ.Asymptotic Expansions in a Parameter?、?1.An existence theorem　Ⅻ-2.Basic estimates?、?3.Proof of Theorem Ⅻ-1-2　Ⅻ-4.A block-diagonalization theorem?、?5.Gevrey asymptotic solutions in a parameter?、?6.Analytic simplification in a parameter　Exercises ⅫChapter ⅩⅢ.Singularities of the Second Kind?、?1.An existence theorem?、?2.Basic estimates?、?3.Proof of Theorem ⅩⅢ-1-2?、?4.A block-diagonalization theorem?、?5.Cyclic vectors (A lemma of P.Deligne)?、?6.The Hukuhara-Turrittin theorem　ⅩⅢ-7.An n-th-order linear differential equation at a singular point of the second kind?、?8.Gevrey property of asymptotic solutions at an irregular singular pointExercises ⅩⅢReferencesIndex

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《常微分方程基礎(chǔ)理論(影印版)》的引進是為了更好地借鑒國外微積分教學(xué)與研究的成功經(jīng)驗，促進我國數(shù)學(xué)教育與研究事業(yè)的發(fā)展，提高高等學(xué)校數(shù)學(xué)教育教學(xué)質(zhì)量，為本科高年級和研究生開展常微分程研究工作提供必要的理論依據(jù)，《常微分方程基礎(chǔ)理論(影印版)》為原版影印，既可供本科高年級和研究生自學(xué)參考，也可做為教材使用。

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